Generalizations of two cardinal inequalities of Hajnal and Juhász

Abstract

A non-empty subset A of a topological space X is called finitely non-Hausdorff if for every non-empty finite subset F of A and every family {Ux:x∈F} of open neighborhoods Ux of x∈F, ⋂{Ux:x∈F}≠∅ and the non-Hausdorff numbernh(X)of X is defined as follows: nh(X):=1+sup{|A|:A⊂X is finitelynon-Hausdorff}. Clearly, if X is a Hausdorff space, then nh(X)=2.We define the non-Hausdorff number of X with respect to singletons, nhs(X), as follows: nhs(X):=1+sup{|clθ({x})|:x∈X}.In 1967, Hajnal and Juhász proved that if X is a Hausdorff space, then: (1) |X|≤2c(X)χ(X) and (2) |X|≤22s(X), where c(X) is the cellularity, χ(X) is the character and s(X) is the spread of X.In this paper we generalize (1) by showing that if X is a topological space, then |X|≤nh(X)c(X)χ(X). Immediate corollary of this result is that (1) holds true for every space X for which nh(X)≤2ω (and even for spaces with nh(X)≤2c(X)χ(X)). This gives an affirmative answer to a question posed by M. Bonanzinga in 2013. A simple example of a T1, first countable, ccc-space X is given such that |X|>2ω and |X|=nh(X)ω=nh(X). This example shows that the upper bound in our inequality is exact and that nh(X) cannot be omitted (in particular, nh(X) cannot always be replaced by 2 even for T1-spaces).In this paper we also generalize (2) by showing that if X is a T1-space, then |X|≤2nhs(X)⋅2s(X). It follows from our result that (2) is true for every T1-space for which nhs(X)≤2s(X). A simple example shows that the presence of the cardinal function nhs(X) in our inequality is essential.